Finite-size scaling for random walks on fractals

Abstract

Random walks are simulated on finite stages of construction of regular fractal lattices. It is proved that the mean-square displacement (RN2) obeys a finite-size scaling hypothesis and the critical exponent nu w is estimated. The efficiency of the method is proved when applied to finitely ramified fractals in which the problem is exactly solvable. nu w is obtained with good accuracy ( approximately=1%) for a class of infinitely ramified fractals, the Sierpinski carpets. The results correct previous estimates based on simulations which did not use finite-size scaling. It is shown that nu w decreases when Dc decreases with very small corrections due to other geometrical properties such as lacunarity. The comparison with estimates of the ideal chain exponent nu c shows that the two problems are not equivalent on these fractals, and that in general nu w> nu c. Estimates of nu w with the same accuracy are obtained on two Sierpinski pastry shells (2<DF<3), where anomalous diffusion is also observed.